# Non-effectively Hyperbolic Characteristics

Chapter

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## Abstract

In this chapter introducing the notion of local and microlocal elementary factorization of *p*, arising from standard techniques of energy integrals we prove that if *p* is of spectral type 1 on *Σ* then *p* always admits a local elementary factorization. On the other hand if *p* is of spectral type 2 even micolocal elementary factorization is not always possible. When *p* is of spectral type 2 near *ρ* we prove that *p* admits a “nice” microlocal factorization near *ρ*, which is also a microlocal elementary factorization at *ρ*, if the cube of some vector field *H*_{ S } annihilates *p* on *Σ* near *ρ*. This factorization is crucial to deriving energy estimates.

## References

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